01 May Question: Please answer the following question. Please show all your working/solutions. In dealing with mac…
Please answer the following question. Please show all your working/solutions.
In dealing with macroeconomic data, it is often informative to express GDP- per-capita in logs. To see the convenience, consider a variable Xt over time. The growth rate of a variable Xt from period t − 1 to period tis given by (Xt − Xt−1)/Xt−1. If we let ∆(Xt) denote the growth rate of the variable Xt from t − 1 to t, we can say that ∆(Xt) is well approximated by log(Xt/Xt−1). Let’s see why.
(a) Show that for small values of z, we must have log(1 + z) ≈ z. (Hint: Use Taylor expansion!). Use this result in the following parts.
(b) Show that log(Xt/Xt−1) ≈ (Xt − Xt−1)/Xt−1 = ∆(Xt). (Hint: add and subtract 1 to Xt/Xt−1 inside the log function.) For the rest of the question, assume ∆(Xt) = log(Xt/Xt−1).
(c) Show ∆(XtYt) = ∆(Xt) + ∆(Yt)
(d) Show ∆(Xt/Yt) = ∆(Xt) − ∆(Yt)
(e) Show ∆(Xt2) = 2∆(Xt)
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